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Pré-Publication, Document De Travail (Preprint/Prepublication) Année : 2023

Critical points of the distance function to a generic submanifold

Charles Arnal
David Cohen-Steiner
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Résumé

In general, the critical points of the distance function d_M to a compact submanifold M ⊂ R^D can be poorly behaved. In this article, we show that this is generically not the case by listing regularity conditions on the critical and µ-critical points of a submanifold and by proving that they are generically satisfied and stable with respect to small C^2 perturbations. More specifically, for any compact abstract manifold M , the set of embeddings i : M → R^D such that the submanifold i(M) satisfies those conditions is open and dense in the Whitney C^2-topology. When those regularity conditions are fulfilled, we prove that the critical points of the distance function to an ε-dense subset of the submanifold (e.g. obtained via some sampling process) are well-behaved. We also provide many examples that showcase how the absence of these conditions can result in pathological cases.
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hal-04471485 , version 1 (21-02-2024)

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Charles Arnal, David Cohen-Steiner, Vincent Divol. Critical points of the distance function to a generic submanifold. 2024. ⟨hal-04471485⟩
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